Film-growth model using level sets

Abstract

A technique for determining a set of surface profiles in a multilayer stack during a fabrication process may be determined using a model of this fabrication process, a geometry of the multilayer stack (such as a pre-defined geometry of the multilayer stack) and a surface profile in the multilayer stack (such as a measured surface profile of the top surface or a bottom surface of the multilayer stack). For example, the model of the fabrication process may be based on a generalized Eikonal equation. In conjunction with deposition rates and/or physical properties of the layers in the multilayer stack, deposition or growth of the multilayer stack may be simulated. Based on the determined set of surface profiles, an acceptance condition of the multilayer stack (such as a multilayer stack in a photo-mask for use in extreme ultra-violet photolithography) may be determined and/or a remedial action may be recommended.

What is claimed is: 
     
       1. A computer-implemented method for determining a set of surface profiles, comprising:
 receiving a surface profile of a surface in a multilayer stack of a reflective photomask, wherein such surface profile was obtained by measuring a top surface of the multilayer stack or a top surface of a substrate underneath such multilayer stack using a microscopy tool to obtain a multi-value grayscale or bitmap representation of such surface; 
 calculating a second surface profile of a second surface in the multilayer stack based on a generalized Eikonal equation, a geometry of the multilayer stack and the surface profile, wherein the second surface is associated with a different layer in the multilayer stack than the surface, and wherein the second surface is adjacent to the surface in the multilayer stack; 
 repeating, one or more times, the calculation operation for additional surfaces in the multilayer stack based on a generalized Eikonal equation, a geometry of the multilayer stack, the surface profile, and surface profiles determined in preceding instances of the calculation operation, thereby determining the set of surface profiles, wherein a given additional surface is associated with a different layer in the multilayer stack than the surface profile and the surface profiles determined in preceding instances of the calculation operation, and wherein the given additional surface is adjacent to another surface in the multilayer stack that is associated with another layer in an immediately preceding instance of the calculation operation, and wherein the surface profile is used as a boundary condition in calculating the set of surface profiles; 
 based on the set of surface profiles, indicating whether the multilayer stack of the photomask will result in a printable defect on a semiconductor wafer during a photolithography process and such photomask requires a remedial action to repair the photomask; and 
 performing the remedial action to repair the photomask if it is indicated that the multilayer stack will result in a printable defect. 
 
     
     
       2. The method of  claim 1 , wherein the surface is a top surface of the multilayer stack. 
     
     
       3. The method of  claim 1 , wherein the calculation operation involves an inverse calculation in which the given additional surface is underneath the other surface. 
     
     
       4. The method of  claim 1 , wherein the surface is a top surface of a substrate underneath the multilayer stack. 
     
     
       5. The method of  claim 1 , wherein the calculation operation involves a forward calculation in which the given additional surface is on top of the other surface. 
     
     
       6. The method of  claim 1 , wherein, prior to calculating the second surface profile, the method further comprises receiving an optical characteristic of the multilayer stack; and
 wherein the calculation operation is further based on the received optical characteristic. 
 
     
     
       7. The method of  claim 1 , wherein the optical characteristic includes information corresponding to a complex index of refraction of the multilayer stack at one wavelength. 
     
     
       8. The method of  claim 1 , wherein the calculation operation is further based on the complex indices of refraction of materials in the multilayer stack proximate to at least one wavelength. 
     
     
       9. The method of  claim 1 , wherein the calculation operation is further based on physical properties of materials in the multilayer stack. 
     
     
       10. The method of  claim 1 , wherein the calculation operation is further based on deposition rates of one or more layers in the multilayer stack. 
     
     
       11. The method of  claim 1 , wherein the calculation operation involves simulating deposition of one or more layers in the multilayer stack. 
     
     
       12. The method of  claim 1 , wherein the surface profile includes a shape that, at a given location on the surface, is associated with more than one value of a vertical displacement. 
     
     
       13. The method of  claim 1 , wherein the surface profile includes a surface roughness. 
     
     
       14. The method of  claim 1 , wherein, after repeating the calculation operation, the method further involves determining an acceptance condition of the multilayer stack based on one or more of the surface profiles in the determined set of surface profiles. 
     
     
       15. The method of  claim 1 , wherein, after determining the acceptance condition, the method further involves providing a remedial action based on the determined acceptance condition. 
     
     
       16. The method of  claim 1 , wherein, at least during the calculation operation, a given surface profile is represented using a level-set function. 
     
     
       17. The method of  claim 1 , wherein the photomask is an extreme ultra-violet photo-mask. 
     
     
       18. The method of  claim 1 , wherein the multilayer stack is associated with a lithographic process. 
     
     
       19. A non-transitory computer-program product for use in conjunction with a computer system, the computer-program product comprising a computer-readable storage medium and a computer-program mechanism embedded therein to determine a set of surface profiles, the computer-program mechanism including:
 instructions for receiving a surface profile of a surface in a multilayer stack of a reflective photomask, wherein such surface profile was obtained by measuring a top surface of the multilayer stack or a top surface of a substrate underneath such multilayer stack using a microscopy tool to obtain a multi-value grayscale or bitmap representation of such surface; 
 instructions for calculating a second surface profile of a second surface in the multilayer stack based on a generalized Eikonal equation, a geometry of the multilayer stack and the surface profile, wherein the second surface is associated with a different layer in the multilayer stack than the surface, and wherein the second surface is adjacent to the surface in the multilayer stack; and 
 instructions for repeating, one or more times, the calculation operation for additional surfaces in the multilayer stack based on a generalized Eikonal equation, a geometry of the multilayer stack, the surface profile, and surface profiles determined in preceding instances of the calculation operation, thereby determining the set of surface profiles, wherein a given additional surface is associated with a different layer in the multilayer stack than the surface profile and the surface profiles determined in preceding instances of the calculation operation, and wherein the given additional surface is adjacent to another surface in the multilayer stack that is associated with another layer in an immediately preceding instance of the calculation operation, and wherein the surface profile is used as a boundary condition in calculating the set of surface profiles; and 
 instructions for, based on the set of surface profiles, indicating whether the multilayer stack of the photomask will result in a printable defect on a semiconductor wafer during a photolithography process and such photomask requires a remedial action to repair the photomask. 
 
     
     
       20. A computer system, comprising:
 at least one processor; 
 at least one memory; and 
 at least one program module, the program module stored in the memory and configured to be executed by the processor to determine a set of surface profiles, the program module including: 
 instructions for receiving a surface profile of a surface in a multilayer stack of a reflective photomask, wherein such surface profile was obtained by measuring a top surface of the multilayer stack or a top surface of a substrate underneath such multilayer stack using a microscopy tool to obtain a multi-value grayscale or bitmap representation of such surface; 
 instructions for calculating a second surface profile of a second surface in the multilayer stack based on a generalized Eikonal equation, a geometry of the multilayer stack and the surface profile, wherein the second surface is associated with a different layer in the multilayer stack than the surface, and wherein the second surface is adjacent to the surface in the multilayer stack; and 
 instructions for repeating, one or more times, the calculation operation for additional surfaces in the multilayer stack based on a generalized Eikonal equation, a geometry of the multilayer stack, the surface profile, and surface profiles determined in preceding instances of the calculation operation, thereby determining the set of surface profiles, wherein a given additional surface is associated with a different layer in the multilayer stack than the surface profile and the surface profiles determined in preceding instances of the calculation operation, and wherein the given additional surface is adjacent to another surface in the multilayer stack that is associated with another layer in an immediately preceding instance of the calculation operation, and wherein the surface profile is used as a boundary condition in calculating the set of surface profiles; and 
 instructions for, based on the set of surface profiles, indicating whether the multilayer stack of the photomask will result in a printable defect on a semiconductor wafer during a photolithography process and such photomask requires a remedial action to repair the photomask.